Low energy proton-proton scattering in effective field theory
Shung-ichi Ando, Jae Won Shin, Chang Ho Hyun, Seung Woo Hong
aa r X i v : . [ nu c l - t h ] N ov October 4, 2007
Low energy proton-proton scatteringin effective field theory
Shung-ichi Ando , Jae Won Shin, Chang Ho Hyun, and Seung Woo Hong Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea
Low energy proton-proton scattering is studied in pionless effective field theory. Em-ploying the dimensional regularization and MS and power divergence subtraction schemesfor loop calculation, we calculate the scattering amplitude in S channel up to next-to-next-to leading order and fix low-energy constants that appear in the amplitude byeffective range parameters. We study regularization scheme and scale dependence in sep-aration of Coulomb interaction from the scattering length and effective range for the S -wave proton-proton scattering.PACS(s): 11.10.Gh, 13.75.Cs. mailto:[email protected] . Introduction Effective field theories (EFTs), which provide us a systematic perturbative schemeand a model-independent calculation method, have become a popular method to studyhadronic reactions with and without external probes at low and intermediate energies.(See, e.g. , Refs. [1, 2, 3, 4, 5] for reviews.) At very low energies, the Coulomb interactionbecomes essential for the study of reactions involving charged particles. The first con-sideration of the Coulomb interaction in a pionless EFT was done by Kong and Ravndal(KR) for low energy S -wave proton-proton ( pp ) scattering [6, 7]. They calculated the pp scattering amplitude up to next-to leading order (NLO). For loop calculations, they em-ployed dimensional regularization with minimum subtraction (MS) scheme and so calledpower divergence subtraction (PDS) scheme suggested by Kaplan, Savage and Wise [8, 9].Then KR estimated a scattering length a ( µ ) for the pp scattering after separating off theCoulomb correction where µ is the scale for dimensional regularization. The leading or-der (LO) result of a ( µ ) was almost infinite at µ = m π where m π is the pion mass [6].In addition, the LO a ( µ ) was highly dependent on the value of µ . Including the NLOcorrection, they obtained a ( µ = m π ) = − . a np in the np channel, a np = − . ± .
009 fm .The value of a ( µ ) deduced after separating the Coulomb and strong interactions isparticularly important in the study of isospin breaking effects in S -wave N N interac-tion [11, 12]. The accurate value of a np is well known as quoted above, while the valuesof the scattering length in the nn channel ( a nn ) and in the pp channel ( a pp ) still haveconsiderable uncertainties.There exists no direct nn scattering experiment because of the lack of free neutrontarget. The values of a nn have been deduced from the experimental data of π − d → nnγ and nd → nnp reactions. Recent publications suggest a nn = − . ± . stat. ) ± . syst. ) ± . th. ) fm from the π − d → nnγ process [13] and a nn = − . ± . − . ± .
35 fm [15] and − . ± . nd → nnp process. Asseen, the values of a nn have significant errors compared to that of a np , and the centervalues do not seem to converge yet. For the pp channel, a very accurate value of the scattering length a C = − . ± . a C = − . ± . pp scatteringdata. It contains however contributions from both strong and electromagnetic interac-tions, and thus we need to disentangle the strong interaction from the electromagneticinteraction. It was pointed out in potential model calculations that there is a consid-erable model dependence in deducing the value of the strong scattering length a pp from a C [19, 21]. Some literature shows a pp = − . ± . a pp = − . ∼ − .
96 fm [22] with uncertainties slightlylarger than those from the potential models.In this work, we employ the pionless EFT [23] including the Coulomb interactionbetween two protons [6, 7] and calculate the pp scattering amplitude with the strong See, e.g., Table VIII in Ref. [10]. Recently, there were proposals to determine the value of a nn more precisely by employing a formalismof EFT, from the π − d → nnγ reaction[17] and neutron-neutron fusion, nn → de − ¯ ν e [18]. N interactions up to next-to-next-to leading order (NNLO). Our main motivation ofthis study is to see how the value of strong scattering length a ( µ = m π ) = − . a ( µ ) on the renormalization schemesand the scale parameter µ .This paper is organized as follows. In Sec. 2 we briefly review the effective rangeformalism for the pp scattering. In Sec. 3 the pionless strong effective Lagrangian up toNNLO is introduced. In Sec. 4 we calculate the S -wave pp scattering amplitude up toNNLO. In Sec. 5, we discuss regularization method and renormalization schemes employedin this work. We renormalize low energy constants (LECs) that appear in the strong N N interaction up to NNLO by effective range parameters employing MS-bar (MS) and PDSschemes and obtain numerical results for the strong scattering length a ( µ ) and strongeffective range r ( µ ). Discussion and conclusions are given in Sec. 6. In Appendix A weshow detailed expressions of the amplitudes in NNLO. Detailed calculations of the loopfunctions employing the dimensional regularization and MS and PDS schemes are givenin Appendix B.
2. Proton-proton scattering in effective range theory
The amplitude of the pp scattering can be decomposed as [24] T = T C + T SC , (1)where T C is the pure Coulomb part and T SC is the “modified” strong amplitude whose S -wave channel we calculate up to NNLO in pionless EFT below.The incoming and outgoing scattering states | Ψ ( ± ) ~p i with the potential ˆ V = ˆ V C + ˆ V S where ˆ V C and ˆ V S are the Coulomb and strong potentials, respectively, are represented interms of the Coulomb states | ψ ( ± ) ~p i as | Ψ ( ± ) ~p i = ∞ X n =0 ( ˆ G ( ± ) C ˆ V S ) n | ψ ( ± ) ~p i , (2)where ˆ G ( ± ) C is the incoming and outgoing Green’s functionˆ G ( ± ) C ( E ) = 1 E − ˆ H − ˆ V C ± iǫ . (3)Here ˆ H = ˆ p /M is the free Hamiltonian of two protons and V C = e / (4 πr ) is the repulsiveCoulomb potential. The Coulomb state | ψ ( ± ) ~p i is obtained by solving the Schr¨odingerequation ( ˆ H − E ) | ψ ( ± ) ~p i = 0 with ˆ H = ˆ H + ˆ V C and thus one has | ψ ( ± ) ~p i = h G ( ± ) C ˆ V C i | ~p i , (4)3here | ~p i is the free wave state. The normalization of | ψ ( ± ) ~p i is such that h ψ ( ± ) ~p | ψ ( ± ) ~q i =(2 π ) δ (3) ( ~p − ~q ). The amplitude T SC is thus obtained by T SC ( ~p ′ , ~p ) = ∞ X n =0 h ψ ( − ) ~p ′ | ˆ V S ( ˆ G (+) C ˆ V S ) n | ψ (+) ~p i . (5)For l = 0 state one has the amplitude T l =0 SC = − πM e iσ p cot δ − ip , (6)where σ l is the Coulomb phase shift σ l = arg Γ(1 + l + iη ) with η = αM/ (2 p ). In theeffective range expansion with the Coulomb interaction, the modified strong phase shift δ l for l = 0 in low energy pp scattering is represented by effective range parameters [25]: C η p cot δ + αM h ( η ) = − a C + 12 r p − P r p + · · · , (7)where C η = 2 πη/ ( e πη −
1) and h ( η ) = Re ψ ( iη ) − ln η . (8) ψ -function is the logarithmic derivative of the Gamma function and Re ψ ( iη ) = η P ∞ ν =1 1 ν ( ν + η ) − C E ; C E is the Euler’s constant, C E = 0 . · · · . Effective range parameters a C , r , P are modified scattering length, effective range, effective volume, respectively.
3. Effective Lagrangian
Pionless effective Lagrangian for strong S -wave N N interaction up to NNLO reads [23,26] L = N † iD + ~D m N N − C h N T P ( S ) a N i † N T P ( S ) a N + 12 C (cid:20) N T P ( S ) a ↔ D N (cid:21) † N T P ( S ) a N + h.c. − C (cid:18) N T P ( S ) a ↔ D N (cid:19) † N T P ( S ) a ↔ D N −
14 ˜ C "(cid:18) N T P ( S ) a ↔ D N (cid:19) † N T P ( S ) a N + h.c. , (9)where D µ is the covariant derivative, ↔ D = ( → D − ← D ), and P ( S ) a is a projection operatorfor the two-nucleon S states, P ( S ) a = √ σ τ τ a . Note that we retain two low energyconstants, C and ˜ C , in NNLO.The strong N N potential is expanded in terms of small momentum asˆ V S = ˆ V + ˆ V + ˆ V + · · · , (10)4here ˆ V , ˆ V , ˆ V are LO, NLO, NNLO potential, respectively, and the matrix elements ofthem are obtained from the Lagrangian in Eq. (9) as h ~q | ˆ V | ~k i = C , (11) h ~q | ˆ V | ~k i = 12 C ( ~q + ~k ) , (12) h ~q | ˆ V | ~k i = 12 C ~q ~k + 14 ˜ C ( ~q + ~k ) , (13)where | ~q i and | ~k i are the intermediate free two-nucleon outgoing and incoming states,respectively: 2 ~q and 2 ~k are the relative momenta for the two protons.In this work we employ the standard counting rules of the strong N N interaction withthe PDS scheme in Refs. [7, 8]. (We will discuss the PDS scheme in detail later.) Forthe strong potential, the LO term C is counted as Q − order, where Q denotes the smallexpansion parameter, and is summed up to an infinite order. The NLO ( C ) and NNLO( C , ˜ C ) terms are counted as Q and Q , respectively, and expanded perturbatively. We treat the Coulomb interaction non-perturbatively using the Green’s function G ( ± ) C inEq. (3). We do not include higher order QED corrections such as the vacuum polarizationeffects reported in Refs. [27].
4. Amplitudes
The amplitude T l =0 SC for the S -wave pp scattering can be written as T l =0 SC = T (0) SC + T (2) SC + T (4) SC + · · · , (14)where T (0) SC , T (2) SC , T (4) SC are LO, NLO, NNLO amplitudes, respectively. By inserting thestrong LO potential ˆ V in Eq. (11) into the amplitude T SC in Eq. (5), we obtain the LOamplitude T (0) SC in terms of loop functions ψ and J : T (0) SC = ∞ X n =0 h ψ ( − ) ~p ′ | ˆ V ( ˆ G (+) C ˆ V ) n | ψ (+) ~p i = C ψ ( p )1 − C J ( p ) , (15)where ψ ( p ) = Z d ~k (2 π ) ψ (+) ~p ( ~k ) = Z d ~k (2 π ) ψ ( − ) ∗ ~p ( ~k ) , (16) J ( p ) = Z d ~k ′ (2 π ) d ~q (2 π ) h ~q | ˆ G (+) C | ~k ′ i . (17)Detailed calculations for the functions ψ and J are given in Appendix B. T (0) SC is sum-mation of the LO strong potential ˆ V , that is, the C terms summed up to the infiniteorder. Note that by changing the LECs C and ˜ C in another linear combination, e.g., C = C ′ + ˜ C ′ and˜ C = C ′ − ˜ C ′ , one can easily see that the term proportional to ˜ C ′ in Eq. (13) vanishes when | ~q | = | ~k | .The ˜ C ′ term, so called off-shell term, is redundant and vanishes when the external legs of the potentialgo on mass-shell. S -wave pp scattering. Gray blobs denote the two-proton Coulomb Green’s function G (+) C , and two nucleon contact vertices denote the strongpotential: the (black) circle and the (red) square represent LO ( C ) and NLO ( C ) vertices,respectively. Small double dots stand for the summation of C terms up to the infiniteorder.At NLO we have four diagrams shown in Fig. 1. They are proportional to C comingfrom V , whereas the C terms are summed up to the infinite order. The NLO amplitudeis written in terms of the loop functions ψ , ψ , J and J as T (2 ,a − d ) SC = C ψ (1 − C J ) [ ψ + C ( ψ J − ψ J )] , (18)with ψ ( p ) = Z d ~k (2 π ) ~k ψ (+) ~p ( ~k ) = Z d ~k (2 π ) ~k ψ ( − ) ∗ ~p ( ~k ) , (19) J ( p ) = Z d ~q (2 π ) d ~q ′ (2 π ) ~q ′ h ~q ′ | ˆ G (+) C | ~q i = Z d ~q (2 π ) d ~q ′ (2 π ) h ~q ′ | ˆ G (+) C | ~q i ~q . (20)Details for ψ and J are given in Appendix B. The NLO amplitude T (2) SC consists of one C and a summation of the C terms up to the infinite order. These LO and NLO amplitudeshave already been obtained by KR in Ref. [7].At NNLO we have three sets of diagrams shown in Figs. 2, 3, and 4. From the first and Figures were prepared using the program JaxoDraw [28] provided by L. Theussl. C . The NNLO amplitudes correspondingto the diagrams in Fig. 2 can be written in terms of the functions ψ , ψ , J and J ,whereas to express the amplitudes for the diagrams in Fig. 3 we need a new function J given below. In the third set of diagrams shown in Fig. 4, we have one NNLO correctionto the amplitude and the NNLO amplitudes for the diagrams in Fig. 4 are proportionalto C or ˜ C . Explicit expressions of the NNLO amplitude from each of the diagrams aregiven in terms of ψ i with i = 0 , , J j with j = 0 , , , T (4 ,a − h ) SC = C − C J ) n ψ J (1 − C J ) + ψ J (1 − C J ) +2 ψ ψ J (1 − C J ) + ψ J ( C J )(3 + C J ) o , (21)where J = Z d ~q (2 π ) d ~q ′ (2 π ) ~q ′ h ~q ′ | ˆ G (+) C | ~q i ~q , (22)whose details are given in Appendix B. 7igure 3: Set 2 of NNLO diagrams. See the caption of Fig. 1 for details.Summing up the amplitudes for the diagrams (i) to (l) in Fig. 4 gives us T (4 ,i − l ) SC = 12 C (1 − C J ) h ψ (1 − C J ) + 2 ψ ψ C J (1 − C J ) + ψ C J i + 12 ˜ C (1 − C J ) [ ψ + C ( ψ J − ψ J )] ψ , (23)where ψ = Z d ~k (2 π ) ψ ( − ) ∗ ~p ( ~k ) ~k = Z d ~k (2 π ) ~k ψ (+) ~p ( ~k ) , (24) J = Z d ~q (2 π ) d ~q ′ (2 π ) ~q ′ h ~q ′ | ˆ G (+) C | ~q i = Z d ~q (2 π ) d ~q ′ (2 π ) h ~q ′ | ˆ G (+) C | ~q i ~q . (25)Calculations of ψ and J are given in Appendix B.
5. Regularization method and renormalization schemes
In the calculation of the loop functions J , J , J and J in Eqs. (17), (20), (22), (25),we encounter infinities and employ the dimensional regularization. We also employ thePDS scheme, suggested by Kaplan, Savage and Wise [8, 9], in which one subtracts thepoles in d = 3 as well as those in d = 4 space-time dimensions so that one obtains anexpected perturbation series in the expansion of the N N potential in Eq. (10) with a8igure 4: Set 3 of NNLO diagrams. Two-proton contact vertices represented by (blue)diamonds denote strong NNLO potential V . See the caption of Fig. 1 for details.given scale µ of the theory. We may check the convergence radius, e.g., for the C term(relative to the C term) in Eq. (10) and have Λ ( µ ) ≡ q C ( µ ) /C ( µ ) = 147 (30.6) MeVwith (without) the PDS terms at µ = m π . Thus a formal convergence of the perturbativeseries of the N N potential in Eq. (10) is improved thanks to the PDS term, and thetheory would be valid up to p ∼ Λ ≃
140 MeV, which is the large scale we assumed inthe pionless theory.The loop functions can be decomposed into a finite term and an infinite one, e.g. J = J fin + J div with J fin = − αM π H ( η ) (the definition of the H ( η ) function is given inAppendix B) and J div = − M π µ + αM π " ǫ − C E + 2 + ln πµ α M ! , (26)where J div is calculated by the dimensional regularization in d = 4 − ǫ dimensions andthe PDS scheme. The first term proportional to the scale µ in the r.h.s. of Eq. (26) is thePDS term and C E is the Euler’s constant mentioned earlier. The scattering amplitudesshould be identical after renormalization even if another renormalization scheme such asoff-shell momentum subtraction scheme discussed in Refs. [26, 29] is employed. However, a ( µ ) and r ( µ ) do depend on the renormalization schemes along with the value of therenormalization scale µ . So, to be consistent with KR, we calculate all the loop functions J i with i = 0 , , , ψ j with j = 0 , , S -wave pp scattering amplitude in terms of the effective range parameters is givenby T l =0 SC = − πM C η e iσ − αM H ( η ) − a C + r p − P r p + · · · , (27)and thus one has − a C + 12 r p − P r p + · · · = αM H ( η ) − πM C η e iσ T l =0 SC = αM H ( η ) − πM ψ T (0) SC − T (2) SC T (0) SC − T (4) SC T (0) SC + T (2) SC T (0) SC + · · · . (28)Comparing the coefficients of the terms proportional to p , p and p in both sides ofEq. (28), we have − a C = − πM ( C − J div + C C (cid:20) αM µ + 12 ( αM ) + C πM
48 ( αM ) µ (cid:21) − C C − C C ! ( αM ) µ ) + O ( α ) , (29)+ 12 r = 4 πM " C C − C C + 13 ˜ C C − C C ! ( αM ) µ + O ( α ) , (30) − P r = 4 πM C C + 12 ˜ C C − C C ! , (31)where we have expanded the r.h.s. of Eqs. (29) and (30) in the order of the fine structureconstant α and neglected the α ( α ) and higher order terms in Eq. (29) (Eq.(30)). Withthree effective range parameters, we cannot determine the four LECs uniquely. There aresome arguments which can constrain the values of C and ˜ C . The C contribution inEq. (29) is of the order of µ , and thus the first ˜ C contribution term is of the lower orderof µ than the C term. For this reason, the ˜ C term is treated as an order higher thanthe C one [23], and consequently the ˜ C term does not appear (at NNLO) in Eq. (29).The other argument is based on the offshell-ness of a term proportional to C − ˜ C [26]. In this case, the term proportional to C − ˜ C is redundant and thus can be removedby assuming C = ˜ C . Because both arguments seem to have some grounds, to checkthe dependency of the results on the values of C and ˜ C we consider the three cases: 1)˜ C = 0 (Ref. [23]), 2) C = ˜ C (Ref. [26]), and 3) C = 0.In Eq. (29) there is the J div term explicitly given in Eq. (26). In the MS scheme usedby KR [6, 7] one subtracts the infinite term αM π ǫ from the J div . One can use another Note that µ is regarded as a large scale, i.e., µ = m π . See the footnote 4. αM π h ǫ − C E + ln(4 π ) i is subtracted. Then we have J MS = − M π µ + αM π (cid:20) ln (cid:18) µ αM (cid:19) + 1 − C E (cid:21) . (32)This leads to a significant subtraction scheme dependence in the scattering length a ( µ ).
6. Numerical results
We may define the strong scattering length and the effective range, respectively, in thezeroth order of α as [7]1 a ( µ ) = 4 πM C ( µ ) + µ , r ( µ ) = 4 πM C ( µ ) C ( µ ) . (33)Inserting the expressions of a ( µ ) and r ( µ ) in Eqs. (33) into Eqs. (29) and (30), we have1 a ( µ ) = " a ( µ ) LO + " a ( µ ) NLO + " a ( µ ) NNLO , (34) r ( µ ) = r − ( αM ) D P r µ + D r µ a C − µ , (35)where " a ( µ ) LO = 1 a C + αM (cid:20) ln (cid:18) µ αM (cid:19) + 1 − C E (cid:21) , (36) " a ( µ ) NLO = − αM r µ − ( αM ) r + π r µ a C − µ , (37) " a ( µ ) NNLO = ( αM ) D P r µ − D r µ r µ a C − µ , (38)and the term linear in αM in Eq. (35) is the NNLO correction to r ( µ ). We have threeset of coefficients, X x (=1 , , = { D , D , D , D } , because of the additional constraintsimposed on the LECs C and ˜ C mentioned before Eq. (32). X = { , , , } correspondsto the case 1) ˜ C = 0, X = { / , , / , / } corresponds to the case 2) ˜ C = C , and X = { / , − , / , / } to the case 3) C = 0. We use the values of effective rangeparameters, a C = − .
82 fm , r = 2 .
78 fm , P ≃ . . (39)We can also have explicit expressions for the LECs C ( µ ), C ( µ ), C ( µ ) and ˜ C ( µ ) fromEqs. (34), (35) and (31) with the constraints for C and ˜ C .In Fig. 5 we plot our result of the strong scattering length a ( µ ) as a function of thescale parameter µ . In the left panel, we plot three curves for the strong scattering length11 a ( µ ) ( f m ) µ (MeV)MS-bar NNLONLOLO -20-15-10-5 0 0 50 100 150 200 250 300 a ( µ ) ( f m ) µ (MeV)NNLO-1NNLO-2NNLO-3 Figure 5: Strong scattering length a ( µ ) [fm] in functions of the scale parameter µ [MeV].In the left panel, a ( µ ) is plotted by the dotted curve, the dashed curve, and the full curve,respectively, for up to LO, NLO, and NNLO. In the right panel, a ( µ ) calculated up toNNLO are plotted for the three different constraints for C and ˜ C , which are explainedin the text. a ( µ ) up to LO, NLO, and NNLO with the constraint ˜ C = 0 (the case 1). We find thatthe NLO correction significantly improves the estimation of a ( µ ), as shown by KR.If one looks into the details more closely, however, there is a quantitative differencein the results of LO and NLO between the MS and MS schemes. The value of the LOscattering length a LO ( µ ) at µ = m π in the MS scheme, which is obtained from Eq. (36), is a MSLO ( µ = m π ) = − .
72 fm. The LO contributions to a ( µ ) can be divided into three terms;1 /a C , the term proportional to a log function and the remaining ones proportional to αM .Evaluating each contribution, we obtain 1 /a C = − . αM ln (cid:16) m π αM (cid:17) = 0 . αM (1 − C E ) = 0 . − in the MS scheme. There is a strong cancellationbetween 1 /a C and the log term which has the order of αM . Consequently 1 /a ( µ = m π )becomes a small value, making its inverse large. In the case of the MS scheme, thecancellation is stronger, having the log term αM ln (cid:16) πm π α M (cid:17) = 0 . αM ( − C E +2) / . − . The cancellation of 1 /a C and the terms proportional to αM makes the value of 1 /a ( µ = m π ) two orders of magnitude smaller than 1 /a C . As aresult, one gets an unrealistically huge scattering length, a MSLO ( µ = m π ) = 738 .
62 fm. Thestrong dependence on the renormalization schemes of the LO contribution to a ( µ ) makesthe EFT result somehow arbitrary.The NLO contribution, Eq. (37) can be divided into terms linear in αM and thoseproportional to ( αM ) . The term linear in αM is comparable in magnitude with the LOcontribution because of the cancellation in LO, as discussed above. More precisely, wehave 1 /a MSLO = − . − αM r µ/ − . − . On the other hand,the numerical value of the contribution proportional to ( αM ) is 0.0015 in units of fm − ,which is about 5% of the terms linear in αM .12LO NNLO-1 NNLO-2 NNLO-3 a ( µ ) − . − . − . − . r ( µ ) — 2 .
73 2.78 2.82Table 1: Numerical estimations (in units of fm) of scattering length a ( µ ) and effectiverange r ( µ ) up to NLO and NNLO without Coulomb effect at µ = 140 MeV.The NNLO contribution is very small, as can be seen from the left and right panels inFig. 5 and Table 1. The reason can be easily found from the expressions for the NNLOterms in Eq. (38). These terms are proportional to ( αM ) . We observed in NLO thatthe ( αM ) term is smaller than the αM order term by an order of magnitude. Themagnitude of ( αM ) terms in NNLO ranges from about 20% to 300% of ( αM ) termsin NLO, depending on the choice of the assumptions on C and ˜ C . Consequently, theNNLO correction to 1 /a ( µ ) is about 1 ∼
6% of the contributions up to NLO, dependingon the constraints of C and ˜ C .In Table 1 we show the estimated values of the strong scattering length a ( µ ) andeffective range r ( µ ) at µ = m π . The NNLO term itself varies by an order of magnitudedepending on the choice of the constraints on C and ˜ C . However, as discussed in aprevious paragraph, its contribution to a ( µ ) is suppressed due to a higher order of αM factor. As a result, the different choice of the constraints on C and ˜ C affects little thefinal result, only a few percents at most. The first correction to r ( µ ) appears at NNLO andis linear in αM , whereas the NLO correction to 1 /a ( µ ) does in the αM order. Contraryto the case of 1 /a ( µ ) where the αM correction plays a crucial role, the αM contributionto r ( µ ) amounts to only about 2% of r . Though the αM order corrections to 1 /a ( µ )and r ( µ ) are of the same order of magnitude, the ( αM ) order contribution to 1 /a C issmaller than that of r by an order of magnitude. Consequently, we have very contrastingbehavior of a ( µ ) and r ( µ ).Thus our results of the strong pp scattering length and effective range up to NNLO,which are estimated by employing the dimensional regularization and the MS and PDSschemes at µ = m π , can be summarized as a ( µ = m π ) = − . ± . , (40) r ( µ = m π ) = 2 . ± .
05 fm , (41)where the error-bars are estimated by the uncertainties due to the constraints on C and˜ C , which could play a similar role to the model dependence in deducing the values of thestrong scattering length a pp and effective range r ,pp in the potential model calculations.
7. Discussion and conclusions We find a minimum point for a ( µ ) at µ ≃ /r ≃
142 MeV, which is very close to the pion mass, µ = m π .
13n this work, we calculated the S -wave pp scattering amplitude up to NNLO in theframework of the pionless EFT. The loop functions were calculated by using the dimen-sional regularization with the MS and PDS schemes. After fixing the LECs by using theeffective range parameters, we estimated the strong scattering length a ( µ ) and the strongeffective range r ( µ ) as functions of µ . The LO contributions to 1 /a ( µ ) are composed of1 /a C and the terms depending on αM arising from the loop diagrams. The smallnessof 1 /a C makes it comparable in magnitude to the αM terms in the same order. Due tothe opposite signs of 1 /a C and the αM terms, furthermore, there is a strong cancellationamong them and thus it makes the LO result for 1 /a ( µ ) suppressed and sensitive to therenormalization schemes. The NLO correction, expanded in powers of αM , begins withthe linear order of αM . The linear αM order correction to a ( µ ) is of the same orderof magnitude as the αM terms in LO, and thus makes the NLO contribution crucial inboth of the MS and MS schemes. The higher αM order terms in NLO, e.g., the termsproportional to ( αM ) are suppressed to a few percents of the leading contribution, sothey can be regarded as a perturbative corrections to both a ( µ ) and r ( µ ). The NNLOterms give us only a fairly minor correction to the results up to NLO. The reason is partlyattributed to the additional order counting of the NNLO terms in powers of αM : The αM order corrections in NNLO begin with ( αM ) . Similar to the ( αM ) contributionin NLO, the terms in NNLO produces small corrections to the results. In conclusion, wecan say that our investigation reveals both bright and shadowy aspects of studying thestrong pp scattering length in EFT. Convergence from NLO to NNLO is satisfactory, butthe LO and NLO results are significantly dependent on the renormalization schemes.Though the quantities of the strong scattering length and effective range from the pp scattering could be regarded as physical quantities, it is unlikely that they can bedetermined unambiguously without the subtraction scheme and renormalization scaledependence within the present framework of EFT. Similar arguments can be found inRefs. [30, 31]. Nevertheless, the strong pp scattering length and effective range are im-portant ingredients for better understanding of the isospin nature of the N N interaction.The problem of the strong pp scattering length may have to be approached at variouslevels, from “first principle calculations” like lattice QCD to more complex systems inwhich a ( µ ) (or equivalently C ( µ )) plays non-trivial roles. Acknowledgments
We thank Yoonbai Kim for a useful comment on our work. S.A. thanks F. Ravndalfor communications. S.A. is supported by Korean Research Foundation and The KoreanFederation of Science and Technology Societies Grant funded by Korean Government(MOEHRD, Basic Research Promotion Fund): the Brain Pool program (052-1-6) andKRF-2006-311-C00271.
Appendix A: Amplitudes in NNLO
In this appendix we present expressions of each of the amplitudes in NNLO in termsof functions, ψ , , and J , , , . Detailed calculations of the ψ and J functions are given14n Appendix B. From the diagram (a) in Fig. 2, we have T (4 ,a ) SC = h ψ ( − ) ~p ′ | ˆ V ˆ G (+) C ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V G (+) C ˆ V | ψ (+) ~p i = C − C J Z d ~q ′ (2 π ) h ψ ( − ) ~p ′ | ˆ V ˆ G (+) C | ~q ′ i Z d ~q (2 π ) h ~q | ˆ G (+) C ˆ V | ψ (+) ~p i = 14 C C − C J ( ψ J + ψ J ) . (42)From the diagrams (b) and (c) in Fig. 2 we have T (4 ,b,c ) SC = h ψ ( − ) ~p ′ | ∞ X n =0 ( ˆ V G (+) C ) n ˆ V ˆ G (+) C ˆ V ˆ G (+) C ∞ X m =0 ( ˆ V G (+) C ) m ˆ V G (+) C ˆ V | ψ (+) ~p i + h ψ ( − ) ~p ′ | ˆ V ˆ G (+) C ∞ X n =0 ( ˆ V G (+) C ) n ˆ V ˆ G (+) C ˆ V ˆ G (+) C ∞ X m =0 ( ˆ V G (+) C ) m ˆ V | ψ (+) ~p i = C ψ (1 − C J ) Z d ~q (2 π ) d ~q ′ (2 π ) h ~q ′ | ˆ G (+) C ˆ V ˆ G (+) C | ~q i× Z d ~k (2 π ) h h ~k | ˆ G (+) C ˆ V | ψ (+) ~p i + h ψ ( − ) ~p ′ | ˆ V ˆ G (+) C | ~k i i = C C ψ J J (1 − C J ) ( ψ J + ψ J ) . (43)From the diagram (d) in Fig. 2, we have T (4 ,d ) SC = h ψ ( − ) ~p ′ | ∞ X l =0 ( ˆ V ˆ G (+) C ) l ˆ V ˆ G (+) C ˆ V ˆ G (+) C ∞ X m =0 ( ˆ V ˆ G (+) C ) m ˆ V ˆ G (+) C ˆ V ˆ G (+) C ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V | ψ (+) ~p i = C ψ (1 − C J ) "Z d ~q ′ (2 π ) d ~q (2 π ) h ~q ′ | ˆ G (+) C ˆ V ˆ G (+) C | ~q i = C C ψ (1 − C J ) J J . (44)From the diagram (e) in Fig. 3 we have T (4 ,e ) SC = h ψ ( − ) ~p ′ | ~V ˆ G (+) C ˆ V | ψ (+) ~p i = C ψ J + ψ J + 2 ψ ψ J ) . (45)From the diagrams (f) and (g) in Fig. 3 we have T (4 ,f,g ) SC = h ψ ( − ) ~p ′ | ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V ˆ G (+) C ˆ V ˆ G (+) C ˆ V | ψ (+) ~p i + h ψ ( − ) ~p ′ | ˆ V ˆ G (+) C ˆ V ˆ G (+) C ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V | ψ (+) ~p i = 12 C C ψ − C J ( ψ J + 2 ψ J J + ψ J J ) . (46)15rom the diagram (h) in Fig. 3 we have T (4 ,h ) SC = h ψ ( − ) ~p ′ | ∞ X m =0 ( ˆ V ˆ G (+) C ) m ˆ V ˆ G (+) C ˆ V ˆ G (+) C ˆ V ˆ G (+) C ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V | ψ (+) ~p i = 14 C C ψ (1 − C J ) (3 J J + J J ) . (47)From the diagram (i) in Fig. 4 we have T (4 ,i ) SC = h ψ ( − ) ~p | ˆ V | ψ (+) ~p i = 12 C ψ + 12 ˜ C ψ ψ . (48)From the diagrams (j) and (k) in Fig. 4 we have T (4 ,j,k ) SC = h ψ ( − ) ~p ′ | ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V ˆ G (+) C ˆ V | ψ (+) ~p i + h ψ ( − ) ~p ′ | ˆ V ˆ G (+) C ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V | ψ (+) ~p i = 12 C ψ − C J h C ψ J + ˜ C ( ψ J + ψ J ) i . (49)From the diagram (l) in Fig. 4 we have T (4 ,l ) SC = h ψ ( − ) ~p ′ | ∞ X m =0 ( ˆ V ˆ G (+) C ) m ˆ V ˆ G (+) C ˆ V ˆ G (+) C ∞ X n =0 ( ˆ V ˆ G (+) C ) n ˆ V | ψ (+) ~p i = 12 C ψ (1 − C J ) h C J + ˜ C J J i . (50) Appendix B: Loop functions
In this appendix, we present ψ functions ( ψ , ψ , ψ ) and J functions ( J , J , J , and J ) employing dimensional regularization and power divergent regularization scheme [7, 8].We first show the calculations of the ψ , ψ , ψ functions in Eqs. (16), (19), (24). ψ : The Fourier transformation of the Coulomb wavefunction ψ ( ± ) ~p ( ~r ) is ψ ( ± ) ~p ( ~k ) = Z d ~rψ ( ± ) ~p ( ~r ) e − i~k · ~r , (51)with ψ ( ± ) ~p ( ~r ) = ∞ X l =0 (2 l + 1) i l R ( ± ) l ( pr ) P l (cos θ ) , (52)where cos θ = ˆ p · ˆ r . One has the relation, ~k · ~r = kr [cos θ cos ˆ θ + sin θ sin ˆ θ cos( φ − ˆ φ )], where ~r and ~k are represented by ( r, θ, φ ) and ( k, ˆ θ, ˆ φ ), respectively. Now we choose ˆ φ = 0 andthen have Z π dφe − ikr sin θ sin ˆ θ cos φ = 2 πJ ( − kr sin θ sin ˆ θ ) , (53)16here J n is a Bessel function and we have used the Bessel’s first integral, J n ( z ) = πi n R π dφe iz cos φ e inφ . Using the relations, Z π dθ sin θP l (cos θ ) J ( − kr sin θ sin ˆ θ ) e − ikr cos θ cos ˆ θ = i l s π − kr P l (cos θ ) J l + ( − kr ) , (54) J l ( − z ) = ( − l J l ( z ), and j l ( z ) = q π z J l + ( z ) where Eq. (54) is obtained from Eq. (15) inRef. [32], we have ψ ( ± ) ~p ( ~k ) = 4 π ∞ X l =0 (2 l + 1) P l (cos ˆ θ ) Z ∞ drr R ( ± ) l ( pr ) j l ( kr ) . (55)Now we calculate ψ by the dimensional regularization. The angular integration willpick up the l = 0 part of the wavefunction, thus we have ψ ( p ) = (cid:18) µ (cid:19) − d Z d d − ~k (2 π ) d − ψ (+) ~p ( ~k )= 4 π (cid:18) µ (cid:19) − d Ω d − (2 π ) d − Z ∞ drr R (+)0 ( pr ) Z ∞ dkk d − j ( kr )= (2 π ) / (cid:18) µ (cid:19) − d Ω d − (2 π ) d − Z ∞ drr − d R (+)0 ( pr ) Z ∞ dρρ d − J ( ρ ) . (56)Using the relation R ∞ dt t α − J ν ( t ) = α − Γ ( (2 − α + ν ) ) Γ (cid:16) α + ν (cid:17) , we have Z ∞ dρρ d − J ( ρ ) = 2 d − Γ (cid:16) d − (cid:17) Γ (cid:16) − d (cid:17) . (57)Furthermore, from Eq. (6.64) of Ref. [24] we have R (+)0 ( pr ) = e iσ C η F (1+ iη, − ipr ) e ipr ,where F ( a ; b ; z ) is the confluent hypergeometric function (or Kummer’s function ofthe first kind). Using the relation, R ∞ e − t t b − F ( a, c ; tz ) = Γ( b ) F ( a, b, c ; z ), where F ( a, b ; c ; z ) is the first hypergeometric function, we have Z ∞ e ipr r − d F (1 + iη, , − ipr ) = Γ(4 − d )( − ip ) d − F (1 + iη, − d,
2; 2) , (58)and thus ψ = (2 π ) / (cid:18) µ (cid:19) − d Ω d − (2 π ) d − e iσ C η Γ(4 − d )( − ip ) d − F (1 + iη, − d,
2; 2)2 d − Γ (cid:16) d − (cid:17) Γ (cid:16) − d (cid:17) . (59)There are no poles at d = 3 and 4 in Eq. (59). Using the relation F (1 + iη, ,
2; 2) = 1and Ω d = 2 π d/ / Γ( d/ ψ = e iσ C η . (60)17 . ψ : ψ ( p ) = (cid:18) µ (cid:19) − d Z d d − ~k (2 π ) d − ψ (+) ~p ( ~k ) ~k = (2 π ) / (cid:18) µ (cid:19) − d Ω d − (2 π ) d − Z ∞ drr − d R (+)0 ( pr ) Z ∞ dρρ d − J ( ρ )= (2 π ) / (cid:18) µ (cid:19) − d Ω d − (2 π ) d − e iσ C η ( − ip ) d − F (1 + iη, − d,
2; 2)2 d − Γ (cid:16) d +12 (cid:17) − d Γ(4 − d )Γ (cid:16) − d (cid:17) . (61)For d = 4 we have ψ = e iσ C η (cid:18) p − α M (cid:19) , (62)where we have used the relation F (1 + iη, − , ,
2) = − η . For d = 3 we have ψ ( d =3)2 = − e iσ C η αM µ − d + · · · , (63)where we have used the relation F (1 + iη, − ,
2; 2) = − iη , and thus we have ψ = e iσ C η (cid:20) p − αM µ −
12 ( αM ) (cid:21) . (64) ψ : ψ = (cid:18) µ (cid:19) − d Z d d − ~k (2 π ) d − ψ (+) ~p ( ~k ) ~k = (2 π ) / (cid:18) µ (cid:19) − d Ω d − (2 π ) d − Z ∞ drr − − d R (+)0 ( pr ) Z ∞ dρρ d + J ( ρ )= (2 π ) / (cid:18) µ (cid:19) − d Ω d − (2 π ) d − e iσ C η ( − ip ) d F (1 + iη, − d,
2; 2) 2 d + Γ (cid:16) d +32 (cid:17) − d )(3 − d ) Γ(4 − d )Γ (cid:16) − d (cid:17) . (65)For d = 4 we have ψ = e iσ C η (cid:18) p − α M p + 124 α M (cid:19) , (66)where we have used the relation F (1 + iη, − ,
2; 2) = (3 − η + 2 η ). For d = 3 wehave ψ d =34 = − e iσ C η αM µ (cid:18) p − α M (cid:19) − d + · · · , (67)18here we have used the relation F (1 + iη, − ,
2; 2) = i η ( − η ). Thus we have ψ = e iσ C η (cid:26) p − (cid:20) αM µ + 56 ( αM ) (cid:21) p + 16 ( αM ) µ + 124 ( αM ) (cid:27) . (68)Now we calculate loop functions J , J , J and J in Eqs. (17), (20), (22), (25) byusing the results of the ψ functions obtained above. J :The function J ( p ) is given by [7] J ( p ) = M Z d ~l (2 π ) πη ( l ) e πη ( l ) − p − l + iǫ , (69)where l = | ~l | . We now separate J into two parts as [7] J ( p ) = J div + J fin , (70)where J div = − M Z d ~l (2 π ) πη ( l ) e πη ( l ) − l , (71) J fin = M Z d ~l (2 π ) πη ( l ) e πη ( l ) − l p p − l + iǫ . (72)As J fin is already calculated in Ref. [7], by changing the parameter x = 2 πη ( l ) andusing the relation Z ∞ dx x ( e x − x + a ) = 12 (cid:20) ln (cid:18) π (cid:19) − πa − ψ (cid:18) π (cid:19)(cid:21) , (73)where ψ is the logarithmic derivative of the Γ-function, we have J fin = − αM π H ( η ) = − αM π h ( η ) − C η M π ( ip ) , (74)where η = αM/ (2 p ), H ( η ) = ψ ( iη ) + iη − ln( iη ), and h ( η ) = ReH ( η ).Next we calculate the divergence part J div in d = 4 − ǫ dimension J div = − M (cid:18) µ (cid:19) − d Z d d − ~q (2 π ) d − πη ( q ) e πη ( q ) − q . (75)Changing the variable x = 2 πη ( q ) = παM/q , we have J div = − M (cid:18) µ (cid:19) − d π ( d − / (2 π ) d − Γ (cid:16) d − (cid:17) ( απM ) d − Z ∞ dx x − d e x − − M (cid:18) µ (cid:19) − d π ( d − / (2 π ) d − Γ (cid:16) d − (cid:17) ( απM ) d − Γ(4 − d ) ζ (4 − d ) , (76)19here we have used the relation Ω d = 2 π d/ / Γ( d/
2) and ζ ( z ) is the Riemann’s zetafunction. For d = 4 − ǫ we have J div = αM π " ǫ − γ + 2 + ln πµ α M ! . (77)We also consider the pole for d = 3, known as the power divergence subtraction (PDS)scheme pole. Using the relation lim s → h ζ ( s ) − s − i = γ , we have the pole at 3-dimension J div = − µM π − d + · · · , (78)and thus we include the PDS counter term and have J div = − M π µ + αM π " ǫ − γ + 2 + ln πµ α M ! . (79) J J = Z d ~q (2 π ) d ~q ′ (2 π ) ~q ′ h ~q ′ | ˆ G (+) C | ~q i = M Z d ~q (2 π ) ψ ( q ) ψ ∗ ( q ) ~p − ~q + iǫ . (80)Using the result of ψ in Eq. (64), we get J ( p ) = (cid:20) p − µαM −
12 ( αM ) (cid:21) J ( p ) − ∆ J , (81)where∆ J = M (cid:18) µ (cid:19) − d Z d d − ~k (2 π ) d − πη ( k ) e πη ( k ) − M (cid:18) µ (cid:19) − d Ω d − (2 π ) d − ( παM ) d − Γ(2 − d ) ζ (2 − d ) . (82)For d = 4 we have ∆ J = 14 πα M ζ ′ ( − , (83)where ζ ′ ( −
2) = − . · · · . For d = 3 we obtain∆ J ( d =3)2 = 148 πα M µ − d + · · · , (84)and by including the PDS counter term we have∆ J = πM
48 ( αM ) µ + πM αM ) ζ ′ ( − . (85) J J = Z d ~q (2 π ) d ~q ′ (2 π ) ~q ′ h ~q ′ | ˆ G (+) C | ~q i ~q = M Z d ~q (2 π ) ψ ( q ) ψ ∗ ( q ) p − q + iǫ = ( p − Ap + A ) J − ( p − A )∆ J − ∆ J , (86)20here A = µαM + ( αM ) , and∆ J = M (cid:18) µ (cid:19) − d Z d d − ~q (2 π ) d − ~q ψ ( q ) ψ ∗ ( q )= M (cid:18) µ (cid:19) − d Ω d − (2 π ) d − ( παM ) d +1 Γ( − d ) ζ ( − d ) . (87)For d = 4 we have ∆ J = 148 π α M ζ ′ ( − , (88)where ζ ′ ( −
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