Двопетлевi РГ-функцiї масивної теорiї поля у довiльних вимiрностях простору
Two-loop Feynman integrals of the massive field theory are explicitly obtained for generic space dimensions . Corresponding renormalization-group functions are expressed in a compact form in terms of Gauss hypergeometric functions. A number of interesting and useful relations is given for these integrals as well as for several special mathematical functions and constants.
Key words: Feynman integrals, special functions, renormalization group, field theory
PACS: 11.10.Kk, 12.38.Bx, 02.30.Gp, 05.70.Jk
Двопетлевi iнтеграли Фейнмана масивної теорiї поля типу розрахованi в явному виглядi для довiльних вимiрностей простору . Вiдповiднi ренормгруповi функцiї записанi у компактному виглядi у термiнах Гаусових гiпергеометричних функцiй. Продемонстровано ряд цiкавих та корисних спiввiдношень для цих iнтегралiв а також для деяких спецiальних математичних функцiй та констант.
Ключовi слова: Iнтеграли Фейнмана, спецiальнi функцiї, ренормалiзацiйна група, теорiя поля
PACS: 11.10.Kk, 12.38.Bx, 02.30.Gp, 05.70.Jk
Two-loop RG functions] Two-loop RG functions of the massive field theory in general dimensions
Condensed Matter Physics, ????, Vol. ?, No ?(??), pp. LABEL:first@page–LABEL:last@page
M. A. Shpot
Institute for Condensed Matter Physics, 79011 Lviv, Ukraine
The dependence of critical exponents of statistical mechanical systems on the physical space dimension is one of the most fundamental features of critical phenomena . Continuous changing of the space dimension has been the basic principle for constructing the famous Wilson-Fisher epsilon expansion  with . However, the Renormalization Group (RG) , which provides the fundamental theoretical basis for such calculations, is nonperturbative in its nature. It does not require that the critical exponents or any other universal quantities must be expanded in some space deviation, coupling constant, inverse number of order-parameter components, etc.
While investigating the critical phenomena, the RG allows, in principle, to work in any setting of interest provided that we are smart enough to find out the appropriate analytical or numerical tools. The -expansion has shown itself as a very useful tool in studying the qualitative features of systems in critical state even in its lowest-order approximations (see, e.g., ). Pushed to higher orders, it allowed to produce very accurate numerical extrapolations for critical exponents of three-dimensional -vector models . An alternative calculational scheme used with comparable success is the so-called -expansion within the massive field theory in fixed dimension put forward by Parisi . This approach has been mainly applied directly in three dimensions to systems of different complexity [7, 8, 9, 10, 11, 12] (for a recent review see ). In  it was used for calculating the exponents of models in two dimensions, and in  — applied to pure and disordered Ising systems in general dimensions .
A natural access to non-integer dimensions provides also the well-known large- expansion [15, 16, 17, 18]. In its general scope, it gives explicit expressions for critical exponents as functions of . These are valid in the whole range between the lower and upper critical dimensionalities, and can be handled analytically. The large- expansion is capable to yield information on dimensional dependencies that are hardly accessible by other means, for example, from the epsilon expansion. Unfortunately, it is very hard to obtain such results in higher orders in , while short series expansions in usually fail to give very accurate numerical estimates for relatively small values of , say . The convergence of truncated expansions has been analyzed on the basis of the field-theoretical approach at in .
The knowledge of dimensional dependencies of critical exponents, or eventually also universal relations of critical amplitudes are of great theoretical and practical interest. The information of this kind broadens out our fundamental knowledge about these main characteristics of the critical behavior. Moreover, the possibility to consider the results, which are not directly related to dimensional expansions, in the vicinity of the upper or lower critical dimensions provides us with useful checks of their correctness or with some new relevant insights.
Let us give several examples. At the beginning of 1980ies Newman and Riedel  used the exact Wegner-Houghton  RG equation and the scaling field method in a study of the pure and dilute Ising models (IM) in general dimensions from the interval .
Pinn et. al.  studied the RG in the hierarchical model in . Some of their results have been given as tables illustrating the dimensional dependencies of critical indices calculated for some sets of discrete non-integer values of . A contact with the -expansion has also been established.
Recently, Ballhausen et. al.  calculated dimensional dependencies of IM critical exponents for by using the first-order derivative expansion of the exact RG equation for the effective average action (for recent reviews see [24, 25]). Their results for and are very similar to that obtained in  and .
In a very recent paper , O’Dwyer and Osborn analyze the Polchinski version  of the exact RG equations [21, 3, 24] and its possible truncations and derivative expansions. This is done both for non-integer dimensions between two and four, and in the epsilon expansion. Such combined approach allowed to better understand the local potential approximation , the derivative expansions used in treatments of the Polchinski equation, and moreover, to suggest an alternative improved truncation of this equation.
Another story happened in the study of the critical behavior of spin models [12, 30, 31, 32, 33]. Apparently, certain controversies in the discussion of these references could be avoided if there would be more well-established information related to the dimensional dependencies of RG functions, fixed points, and critical exponents.
Finally, let us note that there is a big continuous interest in the high-energy physics (HEP) literature in calculating explicit closed-form results for Feynman integrals in general dimensions [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45].
We believe that the results displaying the explicit dependencies of RG functions and physical observables on the space dimension are of fundamental importance. The aim of the present paper is to do a little step in this direction. We return to the article  published some time ago with Holovatch and calculate the closed-form expressions for the two-loop RG functions of the massive field theory, treated in that work numerically, in general dimensions .
2 Two-loop RG functions and Feynman integrals
Let us consider the usual massive [46, 47, 48, 49, 6] -symmetric theory in general dimensions . To second order in renormalized coupling constant , the corresponding and RG functions can be written as [49, 50, 14]
At the fixed point , which is determined by the zero of the function , the function gives the value of the Fisher exponent , while leads to the combination where is the correlation length critical exponent.
The -dependent functions and are defined by the combinations of Feynman diagrams taken at zero external momenta,
The massive two-loop integrals appearing in and are given by
where is a usual Euler’s gamma function (see e.g. ). This integral is absorbed, along with the symmetry factor into the normalization of the dimensionless renormalized coupling constant . With this normalization, the RG function starts with in the three-dimensional theory [51, 7]. On the other hand, since the integral has a pole as approaches 4, in the epsilon expansion the fixed point value will be given by as a consequence of such normalization.
It is easy to see that the folowing relations hold true for the integrals and ,
where the second equality immediately follows from the scaling property of the function .
Here and further on, we parametrize the dimensional dependencies by the usual deviation from the upper critical dimension . Generically, we do not assume that is infinitesimally small.
At the time of writing the paper , the following information was available. At , the integrals and could be easily evaluated analytically as and . At , their numerical values are [52, 50] and . As , both integrals and have poles in , namely and . In the functions and these poles are compensated by division through , so that their expansions start with111 The similar formula given at p. 873 of  is not correct. 
This integral has not been calculated in  since it is "renormalization-dependent" and disappears from the results for the critical exponents . Numerically, . For non-integer , the functions and have been tabulated in  by using numerical computations in (6) and (7). The corresponding graphs have been plotted point by point for . The aim of the following section is to obtain analytically the explicit expressions for these functions in general dimensions .
3 Explicit calculations in general dimensions
We recognize that here we have to deal with an integral representation of the incomplete beta function (see, e.g., , entry 6.6.1). But it is more convenient to express this function in terms of the Gauss hypergeometric function using the relation 6.6.8 of ,
Thus we obtain
However, the argument of the resulting hypergeometric function is not good for further integration over in the limits from to since the factor is singular at and . The situation is substantially improved by using the three-term linear transformation formula 15.38 from  which reads
After some algebraic transformations we obtain for the inner integral from (6)
An essential simplification occurred in the first term due to an appearance of a hypergeometric function with equal nominator and denominator parameters (see, e.g., , entry 15.1.8):
where is the Pochhammer symbol.
Let us substitute the first term of (14) into the outer integral remaining in (6). Recalling the short-hand notation we see that the factor is exactly cancelled by the analogous term in the denominator and, apart from the numerical constants, we remain with
The last equality can be obtained by simply expanding the denominator of the integral via (15) and integrating the resulting series expansion term by term.
The second part of the square brackets in (14) can be handled in a similar fashion. A straightforward calculation employing the Gauss series representation of yields
Now we notice that the hypergeometric function with the argument can be expressed in terms of a similar function with the argument using the the linear transformation formula (see, e.g., , entry 15.3.4)
As , the first hypergeometric function in square brackets reduces to , the second term vanishes due to the explicit factor , and , as it should (see (8)). Despite of the overall factor , there is no singularity at since the combination in square brackets vanishes at . In Appendix we derive the explicit result for in terms of the generalized hypergeometric function .
Here an interesting symmetry of the nominator and denominator parameters in both functions is observed, while they differ one from another by .
Let us consider the second function . The calculation of the double integral (7) is somewhat more involved but the experience of the above calculation allows us to be brief.
While considering the inner integration in (10) we have seen that it is useful to get its result in terms of a Gauss hypergeometric function of the argument , and not which is suggested directly by the integrals over in (6)–(7) and (10). This can be achieved directly on the level of the integral representation. Indeed, changing the initial integration variable via we obtain for the inner integral from (7)
The two terms of are similar to that generated by the inner integration in (see (14)). They can be handled in the same manner inside of the external integral over in (7). According to this decomposition of , we obtain two different contributions to the function . We can write
Similarly as before, using the linear transformation (18), we can reduce all above hypergeometric functions to that with the argument . Further, the repeated use of the contiguous functions’ relations leaves us only with the same two functions which appeared in . Moreover, they build up just the combination defining that function. Finally we arrive at a simple expression of in terms of :
4 Some results in two dimensions
While the integrals and are smooth functions of in the range of our interest, their calculation at requires some care. In order to calculate from (19) or (20), we have to introduce a small deviation from and consider the limits of these expressions. Details of the calculation can be found in appendix A.
We obtain the explicit expression for at in terms of the generalized hypergeometric function :
Using the relation (23) it is easy to obtain the value of :
Searching for some other representations of we have found the relation (details are given in appendix A)
Here is the maximum value of the Clausen’s function (for more information on this and related functions see )
Apparently, the identity (26) does not appear explicitly in the mathematical literature. It supplements the very similar formula
from mathematical tables , entry 126.96.36.199.
5 What can we learn from the HEP literature?
As well as in the condensed-matter physics, the calculation of multi-loop Feynman integrals is an important issue in the high-energy physics literature. Although the Feynman diagrams in four space-time dimensions are of main concern in the HEP theory, very often they are calculated by using the dimensional regularization in arbitrary space-time dimension . It is recognized (see e.g. ) that working in general dimensions is often much easier than directly at .
Actually, the condensed-matter and high-energy physicists are often doing much the same work by pursuing physically different goals. This is, perhaps, most pronounced just in calculations of Feynman integrals. Unfortunately, it happens not very often that authors of these two communities pay enough attention to achievements gained in another one. The aim of the present section is to quote some HEP results related to calculations of the preceding sections. In doing this we show several useful implications of establishing such contacts. By reviewing briefly some HEP references we would like to draw attention of the cond-mat readers to them, as we feel that these are not well known and appreciated in the cond-mat literature.
Some of the results obtained in  are directly related to ours. In particular, the authors of this reference calculate the two-point function . By using the Mellin-Barnes contour integral representation for massive propagators  they derive for an infinite series over linear combinations of functions222 Only quite recently an explicit expression for has been found  in terms of Gauss and Appell hypergeometric functions..
The first two coefficients of the small- expansion of are simply proportional to our functions and . This can be seen by noticing (see (3)-(4)) that and taking into account the relation (5). Thus we get, first,
The last expression for is equivalent to that found in (19) and (20). A direct comparison between (29) and (19)-(20) can be done by applying the contiguous relations for hypergeometric functions with in nominator parameters. This leads to an interesting and non-trivial relation for involved Gauss’ functions differing in as ,
The second check is given by the relation
Now, taking into account the first equality of (29), we reproduce from (31)-(32) the previously obtained expression (23) for in terms of . Thus, the formula (23) is a consequence of recurrent relations that hold true for the small- expansion coefficients of the Feynman integral . In turn, these recurrences follow from the differential equation333For a recent review on applications of differential equations in calculations of Feymnan integrals see . satisfied by the function .
A full expansion of the functions and can be obtained with the help of eq. (4.16) from . Using it we can write
where is the Clausen’s function. This special function appeared readily in Sec. 4, in the calculation of the integrals and in two dimensions. With the first two terms from the sum, we get the expansions
We see that a very special transcendental constant , which is the maximum value of the Clausen’s integral , appears both in the epsilon expansion of near and in the explicit expression of at . This is quite natural because the Feynman integrals of the type considered here obey certain general relations that connect their values in different space dimensions [59, 35, 39, 40].
A simple formula relating the values of Feynman integrals associated with the zero-momentum "sunrise" diagram with arbitrary masses on the lines at and has been found in . In the particular equal-mass case with , this formula reads, in our notation,
Note that the pole terms of the Laurent expansion of are contained only in the last, trivial term, and the value directly enters the finite part of (cf. , Sec. 3-4). This feature directly maps onto the connection between the functions and , which can be written as
It is straightforward to check that the last relation is indeed satisfied by the function given, for instance, by (29).
Let us return to the epsilon expansions (36)–(37). They can be compared with that quoted in (8)–(9). In (36) we see the term of , which is missing in (8). By comparing the terms of we can make the identification
An attempt to calculate this integral with the help of Mathematica  yields an expression in terms of where is the psi function, the logarithmic derivative of the Gamma function (see e.g. ). Thus we get the chain of equalities
The rightmost connection can be also read off from (A.13) of .
6 Concluding remarks
Let us return to the two-loop RG functions defined at the beginning of the Section 2. We see that they can be expressed in terms of a single function of , say
which contains only one non-trivial Gauss’ hypergeometric function. For example, the function is thus given by
Similar expressions can be constructed also for the remaining RG functions and .
Everywhere the dependence on space dimension is given explicitly in a simple parametric form. Obviously, the convenience of such expressions is much better compared to representations in terms of multiple integrals as that in (6)–(7). In principle, this kind of formulas could be used in considering some dimensional expansions not tied to .
Analytical two-loop results are interesting in view of availability of numerical tables of three-loop diagrams of the massive field theory in dimensions . It could be thought of extending explicit calculations to that order.
Appendix A Explicit results in two dimensions
In order to derive the values of the functions and at we have to take and consider the corresponding limits with . Thus we write in (19) as
There exists a simple limit of ,
while each of the hypergeometric functions in has to be expanded to in order to give the finite . To this end, we found it useful to transform these functions via the linear transformation formula (, entry 15.3.3)
Thus we get
At , the integrals and both are equal to
while the difference between them is of order :
Using Mathematica  we get
We did not find in mathematical tables any summation formula for the generalized hypergeometric function appearing in the last two equations. So we looked for the series representations of this function
The middle version of the series appears, indeed, as a partial case of the formula 188.8.131.52 in  expressed in terms of the function and its derivative. But, unfortunately, a simple numerical check at revealed the incorrectness of this entry.
In order to find the right answer, we considered the series expansion
such that and . Its derivative with respect to , multiplied by , is
Differentiating once again we get
Integrating back leads us to
where is the Clausen function . In deriving the last equality we used the entry 184.108.40.206 from  and the fact that the integration constants in the both last indefinite integrations vanish. Here we observe also that the neighboring entry 220.127.116.11 of  involves the series expansion of the same type as the rightmost one in (55), giving, along with 18.104.22.168, the corrected version of the false formula 22.214.171.124 in the came reference.
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Двопетлевi РГ-функцiї масивної теорiї поля у довiльних вимiрностях простору
М .А. Шпот
Iнститут Фiзики Конденсованих Систем НАН України, 79011 Львiв, Україна