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On The Spin-Dependent Potential Between Heavy Quark And Antiquark


NSF-ITP-94-49 TUIMP-TH-94/59 NUHEP-TH-94-10 May, 1994

On The Spin-Dependent Potential Between Heavy Quark And Antiquark

arXiv:hep-ph/9406287v1 11 Jun 1994

Yu-Qi Chen? a,b,d Yu-Ping Kuang? a,c,d and Robert J. Oakes§ e
a China

Center of Advanced Science and Technology (World Laboratory), P. O. Box 8730, Beijing 100080, China

b Institute

for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030 of Modern Physics, Tsinghua University, Beijing 100084, China? of Theoretical Physics, Academia Sinica, Beijing 100080, China

c Institute d Institute e Department

of Physics and Astronomy, Northwestern University, Evanston, IL 60208

Abstract
A new formula for the heavy quark-antiquark spin dependent potential is given by using the techniques developed in the heavy quark e?ective theory. The leading logarithmic quark mass terms emerging from the loop contributions are explicitly extracted and summed up. There is no renormalization scale ambiguity in this new formula. The spin-dependent potential in the new formula is expressed in terms of three independent color-electric and color-magnetic ?eld correlation functions, and it includes both the Eichten-Feinberg’s formula and the one-loop QCD result as special cases.

PACS numbers: 12.38.cy, 12.38.Lg, 12.40.Qq

Typeset Using REVTEX 0

The study of the structure of the spin-dependent (SD) interaction potential between heavy quark and antiquark from QCD is one of the interesting problems in heavy quark physics. So far there are mainly two kinds of approaches in the literature. The ?rst kind of approach starts from the static limit (in?nitely heavy quark limit) and makes relativistic corrections via the 1/m expansion, where m stands for the heavy-quark mass. The formula for the SD potential in this approach was ?rst given by Eichten and Feinberg [1] in which the potential is expressed in terms of certain correlation functions of color-electric and color-magnetic ?elds weighted by the Wilson loop factor. Later Gromes [2] derived an important relation between the spin-independent (SI) and the SD potentials from the Lorentz invariance of the total potential and the correlation functions given in Ref. [1],and it supported the intuitive color-electric ?ux tube picture of color-electric con?nement suggested by Buchm¨ uller [3]. The second kind of approach is to calculate the SD potential from perturbative QCD up to one-loop level and put in the nonperturbative part of the potential by hand [4,5]. In this approach, certain terms containing ln m emerge from the loop contributions. Furthermore, in the case of unequal masses, new structure of the order-1/m2 spin-orbit coupling containing ln m arises in this approach, which is not included in the ?rst kind of approach [4,5]. It seems that there exists a discrepancy between these two kinds of approaches [6]. To understand the essence of this discrepancy and to have a deeper understanding of the ln m-dependence in the SD potential has become an important theoretical problem in heavy quark physics for a quite long time. In a recent paper [7], two of us (Chen and Kuang) derived some general relations between the SI and SD potentials in the spirit of the 1/m expansion by using the technique of the reparameterization invariance [8–10] developed in the heavy quark e?ective theory (HQET), which include the Gromes relation and some new useful relations, and lead to the conclusion that the general structure of the SD potential is the same as that obtained in the second approach. However, such a simple symmetry argument does not concern the problem of the ln m-dependence of the potential. In this paper, we adopt the conventional formulation in the HQET to construct an e?ective QCD Lagrangian including both a heavy quark ?eld and a heavy antiquark ?eld, 1

with which we study the SD interaction potential. The conventional HQET has proved to be very powerful in studying the heavy-light quark systems [11,12,8–10,13,14] and most of the useful techniques developed in it can be applied to the present theory. We emphasize here that we are working at the quark level to study the interaction potential between heavy quarks rather than studying the bound state quarkonia. The reason causing the discrepancy between the two approaches can be easily understood in the e?ective Lagrangian formalism. When an e?ective Lagrangian is constructed by expanding the full theory in terms of 1/m, the ultraviolet-behavior of the theory is changed. Therefore, when the order of the 1/m expansion and the loop integration are exchanged, which is involved in passing from the full theory to the e?ective theory, di?erences such as logarithmic quark mass terms can emerge. In order to reproduce the results of the full theory, one must match the full theory and the e?ective one. With this matching condition these logarithmic quark mass dependent terms can be explicitly extracted in the coe?cients of higher dimensional operators and can be summed up by using the renormalization group equation (RGE). In the original work of Eichten and Feinberg [1] the authors expanded the full heavy fermion propagator in an “external” gluon ?eld A? in terms of 1/m. It is easy to see that each term in their expression corresponds to insertions of operators in the e?ective Lagrangian which contribute to the SD potential up to order 1/m2 , but with only tree level coe?cients. Therefore, the logarithmic terms are not accounted for. This is the essential reason that there is a discrepancy of the quark mass dependence between the Eichten-Feinberg-Gromes (EFG) formula and the one-loop calculation. With this insight, we can improve the EFG formula by summing the logarithmic quark mass terms and establish a consistent picture reconciling these two kinds of approaches in the framework of the e?ective Lagrangian. To this end, we need not only to renormalize the e?ective Lagrangian up to order of 1/m2 , but also to consider the mixing of the nonlocal operators with local four fermion operators when we use the HQET to calculate the heavy quark-antiquark Green’s functions. Then we can follow the methods developed in Ref. [15,1] and obtain a new formula for the spin dependent potential by using the renormalized e?ective Lagrangian. The above results of the two approaches are just two 2

special approximations of our new formula. Moreover, our new formula is independent of the renormalization scale parameter ?, so that it does not su?er from the scale ambiguity as the second approach does. Let us ?rst construct the renormalized e?ective Lagrangian for one heavy quark ?eld up to order of 1/m2 . We start from the full QCD Lagrangian. As in the conventional HQET, we de?ne the heavy quark ?eld hv+ (x) and the heavy antiquark ?eld h′v? (x) related to the original ?eld ψ (x) as [11] hv+ (x) ≡ P+ eimv·x ψ (x), h′v? (x) ≡ P? e?imv·x ψ (x), where v is the velocity of the heavy quark, and P± ≡ (1)

1±/ v . Integrating out the quantum 2 ?uctuation of the quark ?eld, we obtain the heavy quark e?ective Lagrangian Lc [12,10].

After expanding order by order in powers of 1/m, the ?rst three terms are ? v+ (x)iD · vhv+ (x), L 0 = c0 h ? v+ (x) L 1 = c0 c1 h L 2 = c0
2

(2)
2 ?ν

gs ? hv+ (x)(c4 v ν D ? G?ν + ic5 σ ?ν v σ D? Gνσ )hv+ (x) 4m2 gs c0 ? 2 2 G?ν σ ?ν iD · vhv+ (x), ? 2h v+ (x) c6 (iD ) ? ic7 (D · v ) ? c8 4m 2

(iD ) (iD · v ) ? v+ (x) G?ν σ hv+ (x), (3) hv+ (x) ? c0 c2 ? hv+ (x) hv+ (x) + c0 c3 gs h 2m 2m 4m

(4)

a 2 where iD? = i?? ? gAa ? T . The last term in L2 has no contribution in order 1/m due to

the equation of motion. At tree level the ci ’s are all unity. Note that after the expansion the high energy behavior is di?erent from that in the full theory. So the operators in (2)-(4) need to be renormalized and their coe?cients can be determined by matching to the full √ theory. Here c0 corresponds to the wavefunction renormalization constant, and c1 ? c3 have been calculated in Refs. [13] and [14], using the RG summation, Ref. [14] gives αs (?) c1 (?, m) = 1, c2 (?, m) = 3 αs (m)
8 ? 25 9 ? 25

? 2, c3 (?, m) =

αs (?) αs (m)

,

(5)

where ci ≡ ci (?, m). The coe?cients c4 ? c6 can be determined by the reparameterization invariance of Lc which leads to the reparameterization invariance of the renormalized 3

Lagrangian. This is shown as follows: Consider the in?nitesimal velocity transformation v → v + ?v [10]. The in?nitesimal transformation of δ L can be written as δ L = δ T0 + where ? v+ (x)iD · ?vhv+ (x), δ T0 = c0 (1 ? c1 )h and ? v+ (x)iD · ?viD · vhv+ (x) δ T1 = c0 (1 ? 2c2 + c6 ) h / v 1 ? v+ (x)γ ? v ν igG?ν ? c4 ? 1 c )h h (x) + c0 (c3 ? c 2 + 2 2 5 2 v+ / v ? ν 1 ? v+ (x) ? c4 + 1 c )h γ v igG?ν hv+ (x). + c0 (1 ? c2 ? c3 + 2 2 5 2 (7) 1 δ T1 , 2m (6)

(8)

Di?erent terms in (7)-(8) are of di?erent Lorentz structures, and therefore, they should vanish separately. δ T0 = 0 leads to c1 = 1, i.e., the kinetic energy term is not renormalized [9], and δ T1 = 0 gives the two relations c4 = c6 = 2 c2 ? 1 , Using Eq.(5) we obtain αs (?) c4 (?, m) = 6 αs (m)
8 ? 25

c5 = 2 c3 ? 1 .

(9)

? 5,

αs (?) c5 (?, m) = 2 αs (m)

9 ? 25

? 1.

(10)

The e?ective Lagrangian for antiquark ?eld can be obtained by simply replacing v by ?v and h+v (x) by h′?v (x) in the above e?ective Lagrangian. To study the heavy quark-antiquark interaction potential we are going to evaluate the heavy quark-antiquark four point Green’s function to order 1/m2 . In doing this, we need to calculate all the possible order 1/m operator insertions. Then, additional divergences will appear from double insertions of these operators, i.e., these bilocal operators will mix with certain local four-fermion operators. Let us consider a general unequal mass case, e.g. the ?2 with mass m2 , as in the case of the heavy quark Q1 with mass m1 and the antiquark Q 4

c? b. To order 1/m2 , there are only two local dimension 6 color singlet four fermion operators O1 (x) and O2 (x):
2 gs ? v+i (x)σ ?ν hv+j (x)h ?′ v?j (x)σ?ν h′ (x), h v ?i 4m1 m2 2 gs ? v+i (x)σ ?ν hv+i (x)h ?′ v?j (x)σ?ν h′ (x), O2 (x) = h v ?j 4m1 m2

O1 (x) =

(11) (12)

where i, j = 1, 2, · · · , Nc are color indices. Suppose m1 > m2 . The heavy quark antiquark
2 e?ective Lagrangian up to order 1/m2 1 , 1/m1 m2 , and 1/m2 can be constructed in two steps

as follows. Starting from the Lagrangian in the full theory, we ?rst treat Q1 as a heavy quark, and obtain L′ = LQ1 ef f + LQ2 . (13)

We do not need to add new operators in (13) because there are no divergent terms of the form of O1 (x) and O2 (x). Next, we treat Q2 as a heavy antiquark, and obtain L = LQ1 ef f + LQ2 ef f + d1 (?)O1(?) + d2 (?)O2 (?),
′′

(14)

where the last two terms are the two necessary dimension-6 operators with unknown coe?cients d1 (?) and d2 (?), respectively. Now we determine d1 (?) and d2 (?) by using the RGE. It is easy to see that only the magnetic operator insertion in each fermion line will mix with O1 and O2 due to the Lorentz structure. Let us denote the resulting contribution by O0 (x) ≡
2 gs 16m1 m2

? v+ (x)G?ν σ ?ν hv+ (x)h ? ′ (y )Gαβ σ αβ h′ (y ) , d4 yT ? h v? v? Here T ? means time ordering.

(15)

and its coe?cient by d0 (?).

The coe?cients d(?) ≡

(d0 (?), d1 (?), d2 (?)) satisfy the renormalization group equation ? d d (? ) + d (? )γ = 0 , d? (16)

where γ is the anomalous dimension matrix. A straightforward one-loop calculation using the HQET technique gives

5

1 Nc ? ? ? ?Nc ? 8 8? ? g2 ? ?. γ= 2? 0 0 ? ? 0 4π ? ?
? ? ? ?

(17)

0

0

0

Here γ00 = 2γmag . The results γ10 = γ20 = 0 means that local operators are not able to mix with bilocal operators. However, bilocal operators can mix with local operators, and there are two box, two cross, and a “?sh” diagram contributing to γ01 and γ02 . That γ11 = γ12 = γ21 = γ22 = 0 can be understood as follows: The anomalous dimensions are gauge independent. If we take an axial gauge where v · A = 0 gluonic interactions decouple from the fermion ?eld in the zeroth order e?ective Lagrangian (2). The initial condition for the RGE is determined by matching the e?ective theory to the full theory at ? = m2 , i.e. d(m2 ) = ( c3 (m2 , m1 ), 0, 0 ) . With this, the solution of the RGE (16) are
2 25 αs (? ) , d0 (?) = c3 (?, m2 )c3 (?, m1 ) = αs (m1 )αs (m2 ) ? ?9 1 1 αs (m2 ) 25 ? αs (?) 2 d1 (?) = c3 (m2 , m1 )[1 ? c3 (?, m2 )] = 1? 8 8 αs (m1 ) αs (m2 ) 1 d 2 (? ) = ? d 1 (? ). Nc ?
9

(18)

? 18 25

?

?,

(19)

Our e?ective Lagrangian (14) is thus completely determined. Now we apply it to calculate spin-dependent force. We shall take v = (1, 0, 0, 0) and denote hv+ (x), h′v? (x) as h(x), h′ (x) for short. Similar to Ref. [1], we introduce a gauge invariant four-point Green’s function ? ′ (y 2 )Γ ? (x1 )ΓA P (x1 , x2 )h′ (x2 )] |0 , ? B P (y2, y1 )h(y1 )] [h I = 0|T ?[h where P (x, y ) ≡ P exp ig
x y

(20)

dz? A? (z ) is the path-ordered exponential [16,17]. As is argued

0 0 0 in Refs. [15,1], in the limit that the time interval T ≡ (y1 + y2 )/2 ? (x0 1 + x2 )/2 → ∞, with 0 0 0 x0 2 ? x1 and y2 ? y1 ?xed, the limit of I is

I → δAB δ (rx ? ry ) exp[?T ?(r )], 6

(21)

where rx = x1 ? x2 , ry = y1 ? y2 , r = |rx |, and ?(r ) is just the static energy between the quark and antiquark separated by the spatial distance r . Here the appropriate ordering of the limits is that ?rst m → ∞ and then T → ∞, so that the motion of the quark and the antiquark can be treated perturbatively [15,1]. Taking all the operators with dimension higher than 4 in the Lagrangian as the perturbative part, I can be calculated by using standard perturbation theory. In the calculation, the zeroth order full fermion propagator S0 (x, y, A) in the external gluon ?eld A? is used and it is [15,1] S0 (x, y, A) = ?iθ (x0 ? y0 )P (x0 , y0 )δ (x ? y ). ? as Next we de?ne the symbol · · · and I Q(x) ≡ [dA? ]T r P exp ig
C (r,T )

(22)

dz? A? (z ) Q(x)
x∈C

exp(iSY M (A)),

(23)

? P ? ΓI ?)δ (x1 ? y1 )δ (x2 ? y2 ). I ≡ T r (P + Γ
2 ? To order 1/m2 1 , 1/(m1 m2 ) and 1/m1 , I can be expressed as

(24)

? = 1 +i I ? ? g s (? ) 4m2 1 1 m2 1
2

T /2 T /2 T /2 T /2

dz

1 D2 (x1 , z ) ? c3 (?, m1 )gs (?)σ1 · B (x1 , z ) + (1 ? 2) m1

k dz [(c4 (?, m1 )δij ? c5 (?, m1 )i?ijk σ1 ) E i (x1 , z )D j (x1 , z ) + (1 ? 2) T /2 ?T /2 i i dz ′ θ(z ′ ? z )[ (D2 ? c3 (?, m1 )gs (?)σ1 B )(x1 , z )

T /2 ?T /2

dz

(D ?

i i c3 (?, m1 )gs (?)σ1 B )(x1 , z ′ )

1 + (1 ? 2)] ? m1 m2

(25)
T /2 ?T /2

T /2 T /2

dz

dz



j j i i (D2 ? c3 (?, m1 )gs (?)σ1 B )(x1 , z )(D2 ? gs (?)c3 (?, m2 )σ2 B )(x2 , z ′ )

+ where

2 Nc gs (? ) i i 3 T d(?)σ1 σ2 δ (x1 ? x2 ), 2m1 m2

d (? ) = d 1 (? ) + 7

d 2 (? ) . Nc

(26)

Similar to the derivation given in Ref. [1], we obtain the spin-dependent potential V (r ) = V 0 (r ) + S1 S2 + 2 ·L 2 m1 m2 c+ (?, m1 , m2 ) ? 1 2 dV0 (r ) dr

+ c+ (?, m1 , m2 )

dV1 (?, r ) 1 dV2 (?, r ) S1 + S2 + ·Lc+ (?, m1 , m2 ) dr m1 m2 r dr

1 2 1 (S1 · r)(S2 · r) ? 3 S1 · S2 r c3 (?, m1 )c3 (?, m2 )V3 (?, r ) + m1 m2 r2 + + + where 1 c+ (?, m1 , m2 ) = [c3 (?, m1 ) + c3 (?, m2 )], 2 and V0 (r ) ≡ ? lim rk rk
T →∞

(27)

1 1 2 S1 · S2 (c3 (?, m1 )c3 (?, m2 )V4 (?, r ) ? 6Nc gs (? )d (? )δ (r ) 3 m1 m2 S1 S2 1 d[V0 (?, r ) + V1 (?, r )] ? 2 ·Lc? (?, m1 , m2 ) 2 m1 m2 r dr S1 ? S2 1 dV2 (?, r ) ·Lc? (?, m1 , m2 ) , m1 m2 r dr

1 c? (?, m1 , m2 ) = [c3 (?, m1 ) ? c3 (?, m2 )], 2

(28)

ln 1 , T
T /2 ?T /2 T /2 ?T /2

(29) dz
T /2 ?T /2 T /2 ?T /2

1 dV1 (?, r ) ≡ lim ?ijk T →∞ r dr 1 dV2 (?, r ) ≡ lim ?ijk T →∞ r dr

dz ′ dz ′

z′ ? z 2 gs (?)/2 B i(x1 , z )E j (x1 , z ′ ) / 1 , T z′ 2 gs (?)/2 B i (x2 , z )E j (x1 , z ′ ) / 1 , T
T /2 ?T /2

(30)

dz

(31)

[(? ri r ?j ?

δ ij δ ij )V3 (?, r ) + V4 (?, r )] ≡ lim T →∞ 3 3

dz

T /2 ?T /2

dz ′

2 gs (? ) i B (x1 , z )B j (x2 , z ′ ) / 1 . (32) T

d [V0 (r ) + V1 (?, r ) ? V2 (?, r )] = 0. The relation dr c5 (?, m) = 2c3 (?, m) ? 1 in (9) obtained from reparameterization invariance ensures that In Ref. [2], Gromes derived a relation the general relations derived in Ref. [7] are all satis?ed. It also shows that those relations are consistent with each other. Using those relation, the SD potential can be simpli?ed as

8

V (r ) = V 0 (r ) + +

S2 dV2 (?, r ) 1 dV0 (r ) S1 ? + 2 ·L c+ (?, m1 , m2 ) 2 m1 m2 dr 2 dr

S1 + S2 1 dV2 (?, r ) ·Lc+ (?, m1 , m2 ) m1 m2 r dr (33)

1 2 1 (S1 · r)(S2 · r) ? 3 S1 · S2 r c3 (?, m1 )c3 (?, m2 )V3 (?, r ) + m1 m2 r2 + + 1 1 2 S1 · S2 (c3 (?, m1 )c3 (?, m2 )V4 (?, r ) ? 6Nc gs (? )d (? )δ (r ) 3 m1 m2 1 dV2 (?, r ) S2 S1 ? S2 S1 ·L c? (?, m1 , m2 ) ? 2 ·L + . 2 m1 m2 m1 m2 r dr

This is our new formula for the spin-dependent quark-antiquark potential. In Eq. (33), if we take each coe?cient to be its tree level value, i.e., c3 (?, m) = c+ (?, m1 , m2 ) = 1 and c? (?, m1 , m2 ) = d(?) = 0, our result reduces to the EFG formula. Next we compare our result with that in the one-loop QCD calculation. First we see that V5 (?, m1 , m2 ) introduced in Ref. [5] is not an independent function. If we take the oneloop values of c3 (?, m), c± (?, m1 , m2 ), d(?), and then calculate the correlation functions to one-loop, our formula (33) reproduces all the logarithmic mass terms in Refs. [4,5]. In conclusion, the leading logarithmic quark mass terms emerging from the loop contributions are explicitly extracted and summed up by matching the e?ective theory and the full theory and solving the renormalization group equation. The discrepancy appearing in the EFG results and one-loop calculation can then be understood. Our result shows that the e?ective theory can reproduce the full theory beyond tree level in 1/m2 and can be used in calculating the Green’s functions with two heavy quark external lines. One of the authors, Y.-Q. Chen, would like to thank Mark Wise for instructive discussions on this work. He also would like to thank David Kaplan, Lisa Randall and Michael Luke for useful discussions. This work is supported partly by the National Natural Science Foundation of China, the Fundamental Research Foundation of Tsinghua University, and the U.S. Department of Energy, Division of High Energy Physics, under Grant DE-FG0291-ER40684. 9

REFERENCES
? ? § ?

On leave from China Center of Advanced Science and Technology. Email address: ypkuang@ihep.ac.cn Email address: oakes@fnalv.fnal.gov Mailing address. [1] E. Eichten and F. Feinberg, Phys. Rev. D 23 2724 (1982). [2] D. Gromes, Z. Phys.-C 26, 401 (1984). [3] W. Buchm¨ uller, Phys. Lett. B112, 479 (1982). [4] W. Buchm¨ uller, Y. J. Ng, and, S. -H. H. Tye, Phys. Rev. D 24 3003 (1981); S.N. Gupta, S.F. Radford, and W.W. Repko, Phys. Rev. D 26 3305 (1982); S.N. Gupta, and S.F. Radford, Phys. Rev. D 25 3430 (1982). [5] J. Pantaleone, S. -H. H. Tye, and Y. J. Ng, Phys. Rev. D 33 777 (1986). [6] W. Lucha, F.F. Sch¨ orberl, and D. Gromes, Phys. Rep. 200, 127 (1991). [7] Y.-Q. Chen and Y.-P. Kuang, Preprint TUIMP-TH-93/54 and CCAST-93-37. [8] M.J. Dugan, M. Goldenm and B. Grinstein, Phys. Lett. B282 142 (1992). [9] M.E. Luke, A. V. Manohar, Phys. Lett. B286 348 (1992).

[10] Yu-Qi Chen, Phys. Lett. B317 421 (1993). [11] H.D. Politzer and M.B. Wise, Phys. Lett. 206B 681 (1988); 208B 504 (1988); N. Isgur and M.B. Wise, Phys. Lett. 232B 113 (1989); Phys. Lett. 237B 527 (1990); H. Georgi, Phys. Lett. 264B 447 (1991); E. Eichten and B. Hill, Phys. Lett. 233B 427 (1990); B. Grinstein, Nucl. Phys. B339 447 (1990); A.F. Falk, H. Georgi, B. Grinstein, and M.B. Wise, Nucl. Phys. B343 1 (1990); H. Georgi, B. Grinstein, and M.B. Wise, Phys. Lett. 252B 456 (1990); M.E. Luke, Phys. Lett. 252B 447 (1990); A.F. Falk, M. Neubert, and M.E. Luke, Nucl. Phys. B388 363 (1992); [12] T. Mannel, W. Robert, and Z. Ryzak, Nucl. Phys. B368 204 (1992). [13] E. Eichten and B. Hill, Phys. Lett. 234B 511 (1990); [14] A.F. Falk, B. Grinstein, and M.E. Luke, Nucl. Phys. B357 185 (1991). [15] L.S. Brown and W.I. Weisberg, Phys. Rev. D 20 3239 (1979); [16] F.J. Wegner, J. Math. 12 2259 (1971). [17] K. Wilson, Phys. Rev. D 10 2445 (1974).

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